WASHINGTON — President Barack Obama reached out directly on Tuesday to key Republican senators negotiating a sweeping U.S. immigration overhaul and urged them to help craft a bipartisan reform bill as soon as possible.

Facing criticism for not getting personally involved in the delicate process on Capitol Hill, Obama made phone calls to three U.S. senators — Marco Rubio, John McCain and Lindsey Graham — part of a “Gang of Eight” group of Republicans and Democrats working on an immigration package.

Obama stepped in amid tensions with Republicans over the weekend leak of details of a backup immigration reform plan the White House is drafting in case the congressional effort stalls or breaks down.

The Democratic president told the three Republicans that he supports the Senate immigration initiative and “hopes that they can produce a bill as soon as possible that reflects shared core principles on reform,” the White House said in a statement.

But repeating a warning that Obama has already issued in public, the White House also said “he is prepared to submit his own legislation if Congress fails to act.”

Rubio, a rising Cuban-American star in his party and considered crucial to winning conservative backing for any reform deal, had dismissed the White House draft, which included an eight-year path to citizenship for illegal immigrants, as seriously flawed and “dead on arrival in Congress.”

Rubio’s office had denied assertions by the White House over the weekend that it had been engaged with them on the immigration issue, but administration officials insisted again on Tuesday that the sides had held a series of staff-level meetings.

Obama’s calls on Tuesday, including one to Rubio while he was visiting Jerusalem, may help calm the waters for now.

“Senator Rubio appreciated receiving President Obama’s phone call to discuss immigration reform,” said Rubio spokesman Alex Conant. “The senator told the president that he feels good about the ongoing negotiations in the Senate, and is hopeful the final product is something that can pass the Senate with strong bipartisan support.”

Obama emphasized in last week’s State of the Union address the importance of creating a clear path to citizenship for the estimated 11 million immigrants who are in the United States illegally.

Many Republicans stress that there must first be measurable progress in securing the nation’s borders, a condition that will be hard for the president to accept if it drags out the legalization process.

The White House, however, is counting on many Republicans feeling the pressure to move swiftly on immigration reform after they were chastened by Latino voters’ rejection in the November election.

Obama, who met last week with top Democrats involved in the Senate effort, told the Republicans on Tuesday that “commonsense reform needs to include strengthening border security, creating an earned path to citizenship, holding employers accountable, and streamlining legal immigration,” the White House said.

“He thanked the senators for their leadership, and made clear that he and his staff look forward to continuing to work together with their teams to achieve needed reform,” the White House said.